Peter Barath peb at freemail.hu
Sun Dec 2 15:12:11 PST 2007

```Sorry if the thing already has a name.

Let's suppose there is a vote, where the voters are
to chose from a number of numbers. For example,
the membership fee of the club, the minimum age
of application, the size of the office, anything.

Something where we can suppose: If a person prefers
number a over number b, and a > b > c then she will
prefer b over c (because c is even further from a).

Also, if she prefers a over b and a < b < c then she
will prefer b over c (because c is even further from a).

Scenario one. They vote by everyone giving her first
preference then they search the median value. For example:

Let the membership fee be:

------------------------------- (-)
------------------------------- (-)
------------------------------- (+)
------------------------------- (+)
------------------------------- (+)

In this ladder-like scheme I put a sign at the end of each
separating line: the sign is - if there are more votes under
the line, and is + if there are mor votes above the line.
The winner is the 150\$ because there is the change, so more
than half of the voters wants the fee be 150\$ or higher, and
more than half wants it to be 150\$ or lower.

Scenario two. They vote Condorcet, while they keep the
above mentioned convention: if somebody sees 180\$ as the
best option she must prefer 140\$ over 70\$ etc. The
convention says nothing about the preference between, say,
200\$ and 70\$ in this case.

Theorem 1: If in scenario one there is a winner (no ties)
then in scenario two there is a Condorcet-winner and is the
same as the "ladder" winner in scenario one.

Theorem 2: The ladder voting is strategy-free.

I don't waste the space with proofs, they seem pretty obvious.

The theoretical (well, maybe sometimes practical, yes, we
used sometimes to vote about numbers) significance I see
mainly in the second theorem. We don't have many strategy-
free voting methods, only this, the Clarke-tax and the
random methods mentioned usually with the Gibbard-Satterthwaite
theorem: (random vote and runoff between two random candidates)
for more than 2 choseable possibilities.

Peter Barath

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